Abstract

In this work, one provides a justification of the condition that is usually imposed on the parameters of the hypergeometric equation, related to the solutions of the stationary Schrödinger equation for the harmonic oscillator in two-dimensional constant curvature spaces, in order to determine the solutions which are square-integrable. One proves that in case of negative curvature, it is a necessary condition of square integrability and in case of positive curvature, a necessary condition of regularity. The proof is based on the analytic continuation formulas for the hypergeometric function. It is observed also that the same is true in case of a slightly more general potential than the one for harmonic oscillator.

Article outline:I. INTRODUCTIONII. SCHRÖDINGER EQUATIONA. Hypergeometric equation in case of the potential B. Hypergeometric equation in case of the potential C. Square-integrability conditions for the state functions in case of the potential V1D. Square-integrability conditions for the state functions in case of the potential V2III. SQUARE-INTEGRABILITY AND REGULARITY CONDITIONS, AND ANALYTIC CONTINUATION OF THE HYPERGEOMETRIC FUNCTIONA. Analytic continuation for small values of t1. Exceptional casesB. Analytic continuation for large values of 1. Excluded case: α − β = − n, n ∈ ℕ∪{0}IV. DISCUSSION OF RESULTS